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Stitz-Zeager_College_Algebra_e-book

# The vertex is the point on the parabola closest to

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Unformatted text preview: .5.3. Iodine-131 is a commonly used radioactive isotope used to help detect how well the thyroid is functioning. Suppose the decay of Iodine-131 follows the model given in Equation 6.5, and that the half-life10 of Iodine-131 is approximately 8 days. If 5 grams of Iodine-131 is present initially, ﬁnd a function which gives the amount of Iodine-131, A, in grams, t days later. Solution. Since we start with 5 grams initially, Equation 6.5 gives A(t) = 5ekt . Since the half-life is 8 days, it takes 8 days for half of the Iodine-131 to decay, leaving half of it behind. Hence, A(8) = 2.5 which means 5e8k = 2.5. Solving, we get k = 1 ln 1 = − ln(2) ≈ −0.08664, which we can interpret 8 2 8 as a loss of material at a rate of 8.664% daily. Hence, A(t) = 5e− t ln(2) 8 ≈ 5e−0.08664t . We now turn our attention to some more mathematically sophisticated models. One such model is Newton’s Law of Cooling, which we ﬁrst encountered in Example 6.1.2 of Section 6.1. In that example we had a cup of coﬀee cooling from 160◦ F to room temperature 70◦ F according to the formula T (t) = 70 + 90e−0.1t , where t was measured in minutes. In...
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